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HowTo 03: stimulus timing design (hands-on)

• Goal: to design an effective random stimulus presentation

→ end result will be stimulus timing files

→ example: using an event related design, with simple regression to analyze

• Steps:

0. given: experimental parameters (stimuli, # presentations, # TRs, etc.)

1. create random stimulus functions (one for each stimulus type)

2. create ideal reference functions (for each stimulus type)

3. evaluate the stimulus timing design

• Step 0: the (made-up) parameters from HowTo 03 are:

→ 3 stimulus types (the classic experiment: "houses, faces and donuts")

→ presentation order is randomized

→ TR = 1 sec, total number of TRs = 300

→ number of presentations for each stimulus type = 50 (leaving 150 for fixation)

• fixation time should be 30% ~ 50% total scanning time

→ 3 contrasts of interest: each pair-wise comparison

→ refer to directory: AFNI_data1/ht03

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• Step 1: creation of random stimulus functions

→ RSFgen : Random Stimulus Function generator

→ command file: c01.RSFgen

RSFgen -nt 300 -num_stimts 3 \

-nreps 1 50 -nreps 2 50 -nreps 3 50 \

-seed 1234568 -prefix RSF.stim.001.

→ This creates 3 stimulus timing files:

RSF.stim.001.1.1D RSF.stim.001.2.1D RSF.stim.001.3.1D

• Step 2: create ideal response functions (linear regression case)

→ waver: creates waveforms from stimulus timing files

• effectively doing convolution

→ command file: c02.waver

waver -GAM -dt 1.0 -input RSF.stim.001.1.1D

→ this will output (to the terminal window) the ideal response function, by

convolving the Gamma variate function with the stimulus timing function

→ output length allows for stimulus at last TR (= 300 + 13, in this example)

→ use '1dplot' to view these results, command: 1dplot wav.*.1D

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• the first curve (for wav.hrf.001.1.1D) is displayed on the bottom

• x-axis covers 313 seconds, but the graph is extended to a more "round" 325

• y-axis happens to reach 274.5, shortly after 3 consecutive type-2 stimuli

• the peak value for a single curve can be set using the -peak option in waver

→ default peak is 100

• it is worth noting that there are no duplicate curves

• can also use 'waver -one' to put the curves on top of each other

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• Step 3: evaluate the stimulus timing design

→ use '3dDeconvolve -nodata': experimental design evaluation

→ command file: c03.3dDeconvolve

→ command: 3dDeconvolve -nodata \

-nfirst 4 -nlast 299 -polort 1 \-num_stimts 3 \-stim_file 1 "wav.hrf.001.1.1D"\-stim_label 1 "stim_A" \-stim_file 2 "wav.hrf.001.2.1D"\-stim_label 2 "stim_B" \-stim_file 3 "wav.hrf.001.3.1D"\-stim_label 3 "stim_C" \-glt 1 contrasts/contrast_AB \-glt 1 contrasts/contrast_AC \-glt 1 contrasts/contrast_BC

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• Use the 3dDeconvolve output to evaluate the normalized standard deviations of the

contrasts.

• For this HowTo script, the deviations of the GLT's are summed. Other options are valid,

such as summing all values, or just those for the stimuli, or summing squares.

• Output (partial):

Stimulus: stim_A h[ 0] norm. std. dev. = 0.0010Stimulus: stim_B h[ 0] norm. std. dev. = 0.0009Stimulus: stim_C h[ 0] norm. std. dev. = 0.0011General Linear Test: GLT #1 LC[0] norm. std. dev. = 0.0013General Linear Test: GLT #2 LC[0] norm. std. dev. = 0.0012General Linear Test: GLT #3 LC[0] norm. std. dev. = 0.0013

• What does this output mean?

→ What is norm. std. dev.?

→ How does this compare to results using different stimulus timing patterns?

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• Regression Model (General Linear System)

→ Simple Regression Model (one regressor): Y(t) = 0+1t+ r(t)+(t)

• Run 3dDeconvolve with regressor r(t), a time series IRF

→ Deconvolution and Regression Model (one stimulus with a lag of p TR’s):

Y(t) = 0+1t+0f(t)+1f(t-TR)…+pf(t-p*TR)+(t)

• Run 3dDeconvolve with stimulus files (containing 0’s and 1’s)

• Model in Matrix Format: Y = X + → X: design matrix - more rows (TR’s) than columns (baseline parameters + beta weights).

0 1 0 1 0 … p

------------------ ------------------------

1 1 r(0) 1 p fp … f0

1 2 r(1) 1 p+1 fp+1 … f1

. . . . . .

1 N-1 r(N-1) 1 N-1 fN-1 … fN-p-1

→ : random (system) error N(0, 2)

Basics about Regression

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• Solving the Linear System : Y = X + → the basic goal of 3dDeconvolve

→ Least Square Estimate (LSE): making sum of squares of residual

(unknown/unexplained) error ’ minimal Normal equation: (X’X) = X’Y

→ When X is of full rank (all columns are independent), ^ = (X’X)-1X’Y

• Geometric Interpretation:

→ project vector Y onto a space spanned by the regressors (the column vectors of

design matrix X)

→ find shortest distance from Y to X-space

YY

X-SpaceX-Spacerr11

rr22

XX00

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• X matrix examples (very simple - 4 stimulus events, data is perfectly modeled)

→ suppose that we expect the response to a stimulus to look like (0, 2, 1, 0, 0, 0, …)

→ regression: solve Y = * r0 + * r1 (for and )

→ deconvolution: solve Y = * d0 + * d1 + * d2 + * d3 (for )

expected regression deconvolution

response Y

r0 r1 d0 d1 d2 d3

stim 0 10 1 0 1 1 0 0

2 14 1 2 1 0 1 0

1 12 1 1 1 0 0 1

10 1 0 1 0 0 0

stim 0 10 1 0 1 1 0 0

2 14 1 2 1 0 1 0

1 12 1 1 1 0 0 1

stim 0 10 1 0 1 1 0 0

stim 2 0 14 1 2 1 1 1 0

1 2 16 1 3 1 0 1 1

1 12 1 1 1 0 0 1

10 1 0 1 0 0 0

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• X matrix examples (based on modified HowTo 03 script, stimulus #3):→ regression: baseline, linear drift, 1 regressor (ideal response function)→ deconvolution: baseline, linear drift, 8 regressors (lags)

→ decide on appropriate values of: i

Y regression deconvolution - with lags (0-7)

500 1 0 0 1 0 0 0 0 0 0 0 0 0 500 1 1 0 1 1 1 0 0 0 0 0 0 0 500.01 1 2 0.1 1 2 1 1 0 0 0 0 0 0 500.91 1 3 9.1 1 3 0 1 1 0 0 0 0 0 505.60 1 4 56.0 1 4 0 0 1 1 0 0 0 0 513.69 1 5 136.9 1 5 0 0 0 1 1 0 0 0 518.82 1 6 188.2 1 6 0 0 0 0 1 1 0 0 517.42 1 7 174.2 1 7 1 0 0 0 0 1 1 0 512.19 1 8 121.9 1 8 0 1 0 0 0 0 1 1 507.81 1 9 78.1 1 9 0 0 1 0 0 0 0 1 508.06 1 10 80.6 1 10 1 0 0 1 0 0 0 0 510.44 1 11 104.4 1 11 0 1 0 0 1 0 0 0 511.29 1 12 112.9 1 12 0 0 1 0 0 1 0 0 512.49 1 13 124.9 1 13 1 0 0 1 0 0 1 0 513.64 1 14 136.4 1 14 0 1 0 0 1 0 0 1 513.06 1 15 130.6 1 15 0 0 1 0 0 1 0 0 513.32 1 16 133.2 1 16 0 0 0 1 0 0 1 0 513.98 1 17 139.8 1 17 0 0 0 0 1 0 0 1

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• A bad example: see directory AFNI_data1/ht03/bad_stim/c20.bad_stim→ 2 stimuli, 2 lags each→ stimulus 2 happens to follow stimulus 1

baseline linear drift S1 L1 S1 L2 S2 L1 S2 L2 1 0 0 0 0 0 1 1 0 0 0 0 1 2 0 0 0 0 1 3 1 0 0 0 1 4 0 1 1 0 1 5 0 0 0 1 1 6 1 0 0 0 1 7 0 1 1 0 1 8 0 0 0 1 1 9 0 0 0 0 1 10 1 0 0 0 1 11 0 1 1 0 1 12 1 0 0 1 1 13 1 1 1 0 1 14 0 1 1 1 1 15 1 0 0 1 1 16 0 1 1 0 1 17 1 0 0 1 1 18 0 1 1 0 1 19 0 0 0 1

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• Multicollinearity Problem

→ 3dDeconvolve Error: Improper X matrix (cannot invert X’X)

→ X’X is singular (not invertible) ↔ at least one column of X is linearly dependent on the other columns

→ normal equation has no unique solution

→ Simple regression case:

• mistakenly provided at least two identical regressor files, or some inclusive regressors, in 3dDeconvolve

• all regressiors have to be orthogonal (exclusive) with each other

• easy to fix: use 1dplot to diagnose

→ Deconvolution case:

• mistakenly provided at least two identical stimulus files, or some inclusive stimuli, in 3dDeconvolve

easy to fix: use 1dplot to diagnose

• intrinsic problem of experiment design: lack of randomness in the stimuli varying number of lags may or may not help. running RSFgen can help to avoid this

→ see AFNI_data1/ht03/bad_stim/c20.bad_stim

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• Design analysis

→ X’X invertible but cond(X’X) is huge linear system is sensitive difficult to obtain accurate estimates of regressor weights

→ Condition number: a measure of system's sensitivity to numerical computation

• cond(M) = ratio of maximum to minimum eigenvalues of matrix M

• note, 3dDeconvolve can generate both X and (X’X)-1, but not cond()

→ Covariance matrix estimate of regressor coefficients vector :

• s2() = (X’X)-1MSE

• t test for a contrast c’ (including regressor coefficient): t = c’ /sqrt(c’ (X’X)-1c MSE) contrast for condition A only: c = [0 0 1 0 0] contrast between conditions A and B: c = [0 0 1 -1 0] sqrt(c’ (X’X)-1c) in the denominator of the t test indicates the relative stability

and statistical power of the experiment design

• sqrt(c’ (X’X)-1c) = normalized standard deviation of a contrast c’ (including regressor weight) these values are output by 3dDeconvolve

• smaller sqrt(c’ (X’X)-1c) stronger statistical power in t test, and less sensitivity in solving the normal equation of the general linear system

• RSFgen helps find out a good design with relative small sqrt(c’ (X’X)-1c)

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• So are these results good?

stim A: h[ 0] norm. std. dev. = 0.0010stim B: h[ 0] norm. std. dev. = 0.0009stim C: h[ 0] norm. std. dev. = 0.0011GLT #1: LC[0] norm. std. dev. = 0.0013GLT #2: LC[0] norm. std. dev. = 0.0012GLT #3: LC[0] norm. std. dev. = 0.0013

• And repeat… see the script: AFNI_data1/ht03/@stim_analyze→ review the script details:

• 100 iterations, incrementing random seed, storing results in separate files

• only the random number seed changes over the iterations

→ execute the script via command: ./@stim_analyze→ "best" result: iteration 039 gives the minimum sum of the 3 GLTs, among all 100

random designs (see file stim_results/LC_sums)→ the 3dDeconvolve output is in stim_results/3dD.nodata.039

• Recall the Goal: to design an effective random stimulus presentation (while preserving statistical power)

→ Solution: the files stim_results/RSF.stim.039.*.1D

RSF.stim.039.1.1D RSF.stim.039.2.1D RSF.stim.039.3.1D13